`natOrderedRealization`

definition

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module: IndisputableMonolith.Foundation.OrderedLogicRealization
domain: Foundation
line: 28 · github
papers citing: none yet

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IndisputableMonolith.Foundation.OrderedLogicRealization on GitHub at line 28.

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formal source

  25    simp [natCost, h, h']
  26
  27/-- The ordered natural-number realization. -/
  28def natOrderedRealization : LogicRealization where
  29  Carrier := Nat
  30  Cost := Nat
  31  zeroCost := inferInstance
  32  compare := natCost
  33  zero := 0
  34  step := Nat.succ
  35  Orbit := ArithmeticFromLogic.LogicNat
  36  orbitZero := ArithmeticFromLogic.LogicNat.zero
  37  orbitStep := ArithmeticFromLogic.LogicNat.succ
  38  interpret := ArithmeticFromLogic.LogicNat.toNat
  39  interpret_zero := ArithmeticFromLogic.LogicNat.toNat_zero
  40  interpret_step := by
  41    intro n
  42    exact ArithmeticFromLogic.LogicNat.toNat_succ n
  43  orbit_no_confusion := by
  44    intro n h
  45    exact ArithmeticFromLogic.LogicNat.zero_ne_succ n h
  46  orbit_step_injective := ArithmeticFromLogic.LogicNat.succ_injective
  47  orbit_induction := by
  48    intro P h0 hs n
  49    exact ArithmeticFromLogic.LogicNat.induction (motive := P) h0 hs n
  50  orbitEquivLogicNat := Equiv.refl ArithmeticFromLogic.LogicNat
  51  orbitEquiv_zero := rfl
  52  orbitEquiv_step := by intro n; rfl
  53  identity := natCost_self
  54  nonContradiction := natCost_symm
  55  excludedMiddle := True
  56  composition := True
  57  actionInvariant := True
  58  nontrivial := by

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